Additive sparsification of CSPs

Multiplicative cut sparsifiers, introduced by Bencz´ur and Karger [STOC’96], have proved extremely influential and found various applications. Precise characterisations were established for sparsifiability of graphs with other 2-variable predicates on Boolean domains by Filtser and Krauthgamer [SIDMA’17] and non-Boolean domains by Butti and Zivn´y [SIDMA’20]. ˇ Bansal, Svensson and Trevisan [FOCS’19] introduced a weaker notion of sparsification termed “additive sparsification”, which does not require weights on the edges of the graph. In particular, Bansal et al. designed algorithms for additive sparsifiers for cuts in graphs and hypergraphs. As our main result, we establish that all Boolean Constraint Satisfaction Problems (CSPs) admit an additive sparsifier; that is, for every Boolean predicate P : {0, 1} k → {0, 1} of a fixed arity k, we show that CSP(P) admits an additive sparsifier. Under our newly introduced notion of all-but-one sparsification for non-Boolean predicates, we show that CSP(P) admits an additive sparsifier for any predicate P : Dk → {0, 1} of a fixed arity k on an arbitrary finite domain D

The Time for Reconstructing the Attack Graph in DDoS Attacks

Despite their frequency, denial-of-service (DoS\blfootnote{Denial of Service (DoS), Distributed Denial of Service (DDoS), Probabilistic Packet Marking (PPM), coupon collector's problem (CCP)}) and distributed-denial-of-service (DDoS) attacks are difficult to prevent and trace, thus posing a constant threat. One of the main defense techniques is to identify the source of attack by reconstructing the attack graph, and then filter the messages arriving from this source. One of the most common methods for reconstructing the attack graph is Probabilistic Packet Marking (PPM). We focus on edge-sampling, which is the most common method. Here, we study the time, in terms of the number of packets, the victim needs to reconstruct the attack graph when there is a single attacker. This random variable plays an important role in the reconstruction algorithm. Our main result is a determination of the asymptotic distribution and expected value of this time. The process of reconstructing the attack graph is analogous to a version of the well-known coupon collector's problem (with coupons having distinct probabilities). Thus, the results may be used in other applications of this problem.Comment: 31 pages, 5 figures, 1 tabl

Graded Orbital Occupation near Interfaces in a La2NiO4 - La2CuO4 Superlattice

X-ray absorption spectroscopy and resonant soft x-ray reflectivity show a non-uniform distribution of oxygen holes in a La2NiO4 - La2CuO4 (LNO-LCO) superlattice, with excess holes concentrated in the LNO layers. Weak ferromagnetism with Tc = 160 K suggests a coordinated tilting of NiO6 octahedra, similar to that of bulk LNO. Ni d3z2-r2 orbitals within the LNO layers have a spatially variable occupation. This variation of the Ni valence near LNO-LCO interfaces is observed with resonant soft x-ray reflectivity at the Ni L edge, at a reflection suppressed by the symmetry of the structure, and is possible through graded doping with holes, due to oxygen interstitials taken up preferentially by inner LNO layers. Since the density of oxygen atoms in the structure can be smoothly varied with standard procedures, this orbital occupation, robust up to at least 280 K, is tunable.Comment: 11 pages, 8 figure